# Applications of the DA based Normal Form Algorithm on Parameter-Dependent Perturbations

### A Thesis

In partial fulfillment of the requirements for the degree of *Master of Science* from Michigan State University.

### Abstract

Many advanced models in physics use a simpler system as the foundation upon which problem-specific
perturbation terms are added. There are many mathematical methods in perturbation theory
which attempt to solve or at least approximate the solution for the advanced model based on
the solution of the unperturbed system. The analytical approaches have the advantage that their approximation
is an algebraic expression relating all involved quantities in the calculated solution up
to a certain order. However, the complexity of the calculation often increases drastically with the
number of iterations, variables, and parameters considered. On the other hand, the computer-based
numerical approaches are fast once implemented, but their results are only numerical approximations
without a symbolic form. A numerical integrator, for example, takes the initial values and
integrates the ordinary differential equation up to the requested final state and yields the result as
specific numbers. Therefore, no algebraic expression, much less a parameter dependence within
the solution is given. The method presented in this work is based on the differential algebra (DA)
framework, which was first developed to its current extent by Martin Berz *et. al* [3, 4, 5]. The
used DA Normal Form Algorithm is an advancement by Martin Berz from the first arbitrary order
algorithm by Forest, Berz, and Irwin [13], which was based on an DA-Lie approach. Both
structures are already implemented in COSY INFINITY [18] documented in [7, 16, 17]. The result
of the presented method is a numerically calculated algebraic expression of the solution up
to an arbitrary truncation order. This method combines the effectiveness and automatic calculation
of a computer-based numerical approximation and the algebraic relation between the involved
quantities.

A. Weisskopf (2016) (Master's Thesis)

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